Let \(\nu ^*\)  and \(\hat{\nu }_{N}\)  represent the optimal objective function value of (\(Q\) ) and (\(\hat{Q}\) ), respectively. Since (\(\hat{Q}\) ) is a relaxation of (\(Q\) ), and for \(\hat{\gamma } > \gamma \) , \(\hat{\nu }_{N}\)  is a lower bound for \(\nu ^*\) . Indeed, the probability that \(\hat{\nu }_{N}\)  is a lower bound to \(\nu ^*\)  approaches 1 as \(N \rightarrow \infty .\)  If \(\hat{\gamma } < \gamma \) , then, the probability that \(\hat{\nu }_{N}\)  is an upper bound for \(\nu ^*\)  approaches 1 as \(N\rightarrow \infty .\)  Work by [1]} determines the sample size \(N\) , the number of replications \(M\) , and the following approaches to find high confidence upper/lower bounds for (\(Q\) ).
